3.20.96 \(\int \frac {(3+2 x) (1+x+3 x^3)^{2/3}}{x^3 (1+x+x^3)} \, dx\)

Optimal. Leaf size=141 \[ -2^{2/3} \log \left (2^{2/3} \sqrt [3]{3 x^3+x+1}-2 x\right )+2^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{3 x^3+x+1}+x}\right )-\frac {3 \left (3 x^3+x+1\right )^{2/3}}{2 x^2}+\frac {\log \left (2^{2/3} \sqrt [3]{3 x^3+x+1} x+\sqrt [3]{2} \left (3 x^3+x+1\right )^{2/3}+2 x^2\right )}{\sqrt [3]{2}} \]

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Rubi [F]  time = 1.76, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {(3+2 x) \left (1+x+3 x^3\right )^{2/3}}{x^3 \left (1+x+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((3 + 2*x)*(1 + x + 3*x^3)^(2/3))/(x^3*(1 + x + x^3)),x]

[Out]

(3*(1 + x + 3*x^3)^(2/3)*Defer[Int][(((2/(-9 + Sqrt[85]))^(1/3) - ((-9 + Sqrt[85])/2)^(1/3) + 3*x)^(2/3)*(1 +
(2/(-9 + Sqrt[85]))^(2/3) + ((-9 + Sqrt[85])/2)^(2/3) - 3*((2/(-9 + Sqrt[85]))^(1/3) - ((-9 + Sqrt[85])/2)^(1/
3))*x + 9*x^2)^(2/3))/x^3, x])/(((2/(-9 + Sqrt[85]))^(1/3) - ((-9 + Sqrt[85])/2)^(1/3) + 3*x)^(2/3)*(1 + (2/(-
9 + Sqrt[85]))^(2/3) + ((-9 + Sqrt[85])/2)^(2/3) - 3*((2/(-9 + Sqrt[85]))^(1/3) - ((-9 + Sqrt[85])/2)^(1/3))*x
+ 9*x^2)^(2/3)) - ((1 + x + 3*x^3)^(2/3)*Defer[Int][(((2/(-9 + Sqrt[85]))^(1/3) - ((-9 + Sqrt[85])/2)^(1/3) +
3*x)^(2/3)*(1 + (2/(-9 + Sqrt[85]))^(2/3) + ((-9 + Sqrt[85])/2)^(2/3) - 3*((2/(-9 + Sqrt[85]))^(1/3) - ((-9 +
Sqrt[85])/2)^(1/3))*x + 9*x^2)^(2/3))/x^2, x])/(((2/(-9 + Sqrt[85]))^(1/3) - ((-9 + Sqrt[85])/2)^(1/3) + 3*x)
^(2/3)*(1 + (2/(-9 + Sqrt[85]))^(2/3) + ((-9 + Sqrt[85])/2)^(2/3) - 3*((2/(-9 + Sqrt[85]))^(1/3) - ((-9 + Sqrt
[85])/2)^(1/3))*x + 9*x^2)^(2/3)) + ((1 + x + 3*x^3)^(2/3)*Defer[Int][(((2/(-9 + Sqrt[85]))^(1/3) - ((-9 + Sqr
t[85])/2)^(1/3) + 3*x)^(2/3)*(1 + (2/(-9 + Sqrt[85]))^(2/3) + ((-9 + Sqrt[85])/2)^(2/3) - 3*((2/(-9 + Sqrt[85]
))^(1/3) - ((-9 + Sqrt[85])/2)^(1/3))*x + 9*x^2)^(2/3))/x, x])/(((2/(-9 + Sqrt[85]))^(1/3) - ((-9 + Sqrt[85])/
2)^(1/3) + 3*x)^(2/3)*(1 + (2/(-9 + Sqrt[85]))^(2/3) + ((-9 + Sqrt[85])/2)^(2/3) - 3*((2/(-9 + Sqrt[85]))^(1/3
) - ((-9 + Sqrt[85])/2)^(1/3))*x + 9*x^2)^(2/3)) - 4*Defer[Int][(1 + x + 3*x^3)^(2/3)/(1 + x + x^3), x] + Defe
r[Int][(x*(1 + x + 3*x^3)^(2/3))/(1 + x + x^3), x] - Defer[Int][(x^2*(1 + x + 3*x^3)^(2/3))/(1 + x + x^3), x]

Rubi steps

\begin {align*} \int \frac {(3+2 x) \left (1+x+3 x^3\right )^{2/3}}{x^3 \left (1+x+x^3\right )} \, dx &=\int \left (\frac {3 \left (1+x+3 x^3\right )^{2/3}}{x^3}-\frac {\left (1+x+3 x^3\right )^{2/3}}{x^2}+\frac {\left (1+x+3 x^3\right )^{2/3}}{x}+\frac {\left (-4+x-x^2\right ) \left (1+x+3 x^3\right )^{2/3}}{1+x+x^3}\right ) \, dx\\ &=3 \int \frac {\left (1+x+3 x^3\right )^{2/3}}{x^3} \, dx-\int \frac {\left (1+x+3 x^3\right )^{2/3}}{x^2} \, dx+\int \frac {\left (1+x+3 x^3\right )^{2/3}}{x} \, dx+\int \frac {\left (-4+x-x^2\right ) \left (1+x+3 x^3\right )^{2/3}}{1+x+x^3} \, dx\\ &=-\frac {\left (1+x+3 x^3\right )^{2/3} \int \frac {\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}{x^2} \, dx}{\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}+\frac {\left (1+x+3 x^3\right )^{2/3} \int \frac {\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}{x} \, dx}{\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}+\frac {\left (3 \left (1+x+3 x^3\right )^{2/3}\right ) \int \frac {\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}{x^3} \, dx}{\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}+\int \left (-\frac {4 \left (1+x+3 x^3\right )^{2/3}}{1+x+x^3}+\frac {x \left (1+x+3 x^3\right )^{2/3}}{1+x+x^3}-\frac {x^2 \left (1+x+3 x^3\right )^{2/3}}{1+x+x^3}\right ) \, dx\\ &=-\left (4 \int \frac {\left (1+x+3 x^3\right )^{2/3}}{1+x+x^3} \, dx\right )-\frac {\left (1+x+3 x^3\right )^{2/3} \int \frac {\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}{x^2} \, dx}{\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}+\frac {\left (1+x+3 x^3\right )^{2/3} \int \frac {\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}{x} \, dx}{\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}+\frac {\left (3 \left (1+x+3 x^3\right )^{2/3}\right ) \int \frac {\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}{x^3} \, dx}{\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}+\int \frac {x \left (1+x+3 x^3\right )^{2/3}}{1+x+x^3} \, dx-\int \frac {x^2 \left (1+x+3 x^3\right )^{2/3}}{1+x+x^3} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.32, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(3+2 x) \left (1+x+3 x^3\right )^{2/3}}{x^3 \left (1+x+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((3 + 2*x)*(1 + x + 3*x^3)^(2/3))/(x^3*(1 + x + x^3)),x]

[Out]

Integrate[((3 + 2*x)*(1 + x + 3*x^3)^(2/3))/(x^3*(1 + x + x^3)), x]

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IntegrateAlgebraic [A]  time = 0.46, size = 141, normalized size = 1.00 \begin {gather*} -\frac {3 \left (1+x+3 x^3\right )^{2/3}}{2 x^2}+2^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+x+3 x^3}}\right )-2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{1+x+3 x^3}\right )+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{1+x+3 x^3}+\sqrt [3]{2} \left (1+x+3 x^3\right )^{2/3}\right )}{\sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((3 + 2*x)*(1 + x + 3*x^3)^(2/3))/(x^3*(1 + x + x^3)),x]

[Out]

(-3*(1 + x + 3*x^3)^(2/3))/(2*x^2) + 2^(2/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(1 + x + 3*x^3)^(1/3))] -
2^(2/3)*Log[-2*x + 2^(2/3)*(1 + x + 3*x^3)^(1/3)] + Log[2*x^2 + 2^(2/3)*x*(1 + x + 3*x^3)^(1/3) + 2^(1/3)*(1
+ x + 3*x^3)^(2/3)]/2^(1/3)

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fricas [B]  time = 11.63, size = 380, normalized size = 2.70 \begin {gather*} \frac {2 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} x^{2} \arctan \left (\frac {3 \, \sqrt {3} \left (-4\right )^{\frac {2}{3}} {\left (7 \, x^{7} + 8 \, x^{5} + 8 \, x^{4} + x^{3} + 2 \, x^{2} + x\right )} {\left (3 \, x^{3} + x + 1\right )}^{\frac {2}{3}} - 6 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (55 \, x^{8} + 20 \, x^{6} + 20 \, x^{5} + x^{4} + 2 \, x^{3} + x^{2}\right )} {\left (3 \, x^{3} + x + 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (433 \, x^{9} + 255 \, x^{7} + 255 \, x^{6} + 39 \, x^{5} + 78 \, x^{4} + 40 \, x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}}{3 \, {\left (323 \, x^{9} + 105 \, x^{7} + 105 \, x^{6} - 3 \, x^{5} - 6 \, x^{4} - 4 \, x^{3} - 3 \, x^{2} - 3 \, x - 1\right )}}\right ) + 2 \, \left (-4\right )^{\frac {1}{3}} x^{2} \log \left (\frac {3 \, \left (-4\right )^{\frac {2}{3}} {\left (3 \, x^{3} + x + 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (3 \, x^{3} + x + 1\right )}^{\frac {2}{3}} x - \left (-4\right )^{\frac {1}{3}} {\left (x^{3} + x + 1\right )}}{x^{3} + x + 1}\right ) - \left (-4\right )^{\frac {1}{3}} x^{2} \log \left (-\frac {6 \, \left (-4\right )^{\frac {1}{3}} {\left (7 \, x^{4} + x^{2} + x\right )} {\left (3 \, x^{3} + x + 1\right )}^{\frac {2}{3}} - \left (-4\right )^{\frac {2}{3}} {\left (55 \, x^{6} + 20 \, x^{4} + 20 \, x^{3} + x^{2} + 2 \, x + 1\right )} - 24 \, {\left (4 \, x^{5} + x^{3} + x^{2}\right )} {\left (3 \, x^{3} + x + 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{4} + 2 \, x^{3} + x^{2} + 2 \, x + 1}\right ) - 9 \, {\left (3 \, x^{3} + x + 1\right )}^{\frac {2}{3}}}{6 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+2*x)*(3*x^3+x+1)^(2/3)/x^3/(x^3+x+1),x, algorithm="fricas")

[Out]

1/6*(2*sqrt(3)*(-4)^(1/3)*x^2*arctan(1/3*(3*sqrt(3)*(-4)^(2/3)*(7*x^7 + 8*x^5 + 8*x^4 + x^3 + 2*x^2 + x)*(3*x^
3 + x + 1)^(2/3) - 6*sqrt(3)*(-4)^(1/3)*(55*x^8 + 20*x^6 + 20*x^5 + x^4 + 2*x^3 + x^2)*(3*x^3 + x + 1)^(1/3) +
sqrt(3)*(433*x^9 + 255*x^7 + 255*x^6 + 39*x^5 + 78*x^4 + 40*x^3 + 3*x^2 + 3*x + 1))/(323*x^9 + 105*x^7 + 105*
x^6 - 3*x^5 - 6*x^4 - 4*x^3 - 3*x^2 - 3*x - 1)) + 2*(-4)^(1/3)*x^2*log((3*(-4)^(2/3)*(3*x^3 + x + 1)^(1/3)*x^2
- 6*(3*x^3 + x + 1)^(2/3)*x - (-4)^(1/3)*(x^3 + x + 1))/(x^3 + x + 1)) - (-4)^(1/3)*x^2*log(-(6*(-4)^(1/3)*(7
*x^4 + x^2 + x)*(3*x^3 + x + 1)^(2/3) - (-4)^(2/3)*(55*x^6 + 20*x^4 + 20*x^3 + x^2 + 2*x + 1) - 24*(4*x^5 + x^
3 + x^2)*(3*x^3 + x + 1)^(1/3))/(x^6 + 2*x^4 + 2*x^3 + x^2 + 2*x + 1)) - 9*(3*x^3 + x + 1)^(2/3))/x^2

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (3 \, x^{3} + x + 1\right )}^{\frac {2}{3}} {\left (2 \, x + 3\right )}}{{\left (x^{3} + x + 1\right )} x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+2*x)*(3*x^3+x+1)^(2/3)/x^3/(x^3+x+1),x, algorithm="giac")

[Out]

integrate((3*x^3 + x + 1)^(2/3)*(2*x + 3)/((x^3 + x + 1)*x^3), x)

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maple [C]  time = 19.56, size = 695, normalized size = 4.93

method	result	size
risch	\(-\frac {3 \left (3 x^{3}+x +1\right )^{\frac {2}{3}}}{2 x^{2}}+\RootOf \left (\textit {\_Z}^{3}+4\right ) \ln \left (-\frac {\RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{3}+5 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{3}+4 \left (3 x^{3}+x +1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x +4 \left (3 x^{3}+x +1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{2}+\left (3 x^{3}+x +1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{2}-5 \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}-25 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) x^{3}-7 \left (3 x^{3}+x +1\right )^{\frac {2}{3}} x -\RootOf \left (\textit {\_Z}^{3}+4\right ) x -5 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) x -\RootOf \left (\textit {\_Z}^{3}+4\right )-5 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right )}{x^{3}+x +1}\right )+2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) \ln \left (\frac {5 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{3}+4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{3}+8 \left (3 x^{3}+x +1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x +8 \left (3 x^{3}+x +1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{2}+14 \left (3 x^{3}+x +1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{2}+15 \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}+12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) x^{3}-2 \left (3 x^{3}+x +1\right )^{\frac {2}{3}} x +5 \RootOf \left (\textit {\_Z}^{3}+4\right ) x +4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) x +5 \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right )}{x^{3}+x +1}\right )\)	\(695\)
trager	\(\text {Expression too large to display}\)	\(1455\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+2*x)*(3*x^3+x+1)^(2/3)/x^3/(x^3+x+1),x,method=_RETURNVERBOSE)

[Out]

-3/2*(3*x^3+x+1)^(2/3)/x^2+RootOf(_Z^3+4)*ln(-(RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*RootOf(_Z^3
+4)^3*x^3+5*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)^2*RootOf(_Z^3+4)^2*x^3+4*(3*x^3+x+1)^(2/3)*Roo
tOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*RootOf(_Z^3+4)^2*x+4*(3*x^3+x+1)^(1/3)*RootOf(_Z^3+4)^2*x^2+(
3*x^3+x+1)^(1/3)*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*RootOf(_Z^3+4)*x^2-5*RootOf(_Z^3+4)*x^3-2
5*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*x^3-7*(3*x^3+x+1)^(2/3)*x-RootOf(_Z^3+4)*x-5*RootOf(Root
Of(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*x-RootOf(_Z^3+4)-5*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2
))/(x^3+x+1))+2*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*ln((5*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(
_Z^3+4)+4*_Z^2)*RootOf(_Z^3+4)^3*x^3+4*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)^2*RootOf(_Z^3+4)^2*
x^3+8*(3*x^3+x+1)^(2/3)*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*RootOf(_Z^3+4)^2*x+8*(3*x^3+x+1)^(
1/3)*RootOf(_Z^3+4)^2*x^2+14*(3*x^3+x+1)^(1/3)*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*RootOf(_Z^3
+4)*x^2+15*RootOf(_Z^3+4)*x^3+12*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*x^3-2*(3*x^3+x+1)^(2/3)*x
+5*RootOf(_Z^3+4)*x+4*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*x+5*RootOf(_Z^3+4)+4*RootOf(RootOf(_
Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2))/(x^3+x+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (3 \, x^{3} + x + 1\right )}^{\frac {2}{3}} {\left (2 \, x + 3\right )}}{{\left (x^{3} + x + 1\right )} x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+2*x)*(3*x^3+x+1)^(2/3)/x^3/(x^3+x+1),x, algorithm="maxima")

[Out]

integrate((3*x^3 + x + 1)^(2/3)*(2*x + 3)/((x^3 + x + 1)*x^3), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (2\,x+3\right )\,{\left (3\,x^3+x+1\right )}^{2/3}}{x^3\,\left (x^3+x+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x + 3)*(x + 3*x^3 + 1)^(2/3))/(x^3*(x + x^3 + 1)),x)

[Out]

int(((2*x + 3)*(x + 3*x^3 + 1)^(2/3))/(x^3*(x + x^3 + 1)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+2*x)*(3*x**3+x+1)**(2/3)/x**3/(x**3+x+1),x)

[Out]

Timed out

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